Maxwell balance for a scalar reaction-diffusion front
= Maxwell balance for a scalar reaction-diffusion front
{c}
A stationary <front solution> of $u_t=u_{xx}+f(u)$ connecting equilibria $u_-$ and $u_+$ requires $\int_{u_-}^{u_+}f(s)\,ds=0$. Multiply the stationary equation by $u_x$ and integrate: the derivative term has zero endpoint contribution. This equal-potential condition is the Maxwell balance. For $f(u)=\mu u+\alpha u^3-u^5$, a front from zero to a nonzero constant requires $u_+^2=3\alpha/4$ and $\mu=-3\alpha^2/16$.